The theory you explained is the most supported, however there is other two.

It may be derived from the Pillars of Hercules on the Spanish Coat of arms of the Spanish dollar. These Pillars of Hercules on the silver Spanish dollar coins take the form of two vertical bars (||) and a swinging cloth band in the shape of an "S".

Yet another explanation suggests that the dollar sign was formed from the capital letters U and S written or printed one on top of the other. This theory, popularized by novelist Ayn Rand in Atlas Shrugged, does not consider the fact that the symbol was already in use before the formation of the United States

Source: wikipedia